Lecture 2, November 16: A Classical Model (Galí, Chapter 2)

Transcription

1 MakØk3, Fall 2010 (blok 2) Business cycles and monetary stabilization policies Henrik Jensen Department of Economics University of Copenhagen Lecture 2, November 16: A Classical Model (Galí, Chapter 2) c 2010 Henrik Jensen. This document may be reproduced for educational and research purposes, as long as the copies contain this notice and are retained for personal use or distributed free.

2 Introductory remarks The standard neoclassical model for (exogenous) economic growth is the simple Solow model featuring a xed savings rate (no microfoundations) When extended with optimizing savings behavior, we get the Ramsey model Heavily applied model in macroeconomics Models have no role for money and monetary policy Purpose of this lecture is to introduce money in such a classical model with microfoundations How will money and monetary policy a ect the economy? Does this type of workhorse model give implications and predictions that looks like the real world (so it could be a useful tool)? While the end result is not uplifting, we luckily learn a lot of material that are extremely useful in next lectures 1

3 A basic classical model Goods market Demand side: Households consume (based on utility maximization) Supply side: Firms produce consumption good (maximize pro ts under perfect competition) Labor market Demand side: Firms hire labor (maximize pro ts under perfect competition) Supply side: Households supply labor (based on utility maximization) Financial markets Households optimally invest in a one-period risk-less bond Households also hold money. Why? We just assume it for now All prices are perfectly exible 2

4 Household behavior Household maximize expected utility X 1 E 0 t=0 t U (C t ; N t ) ; 0 < < 1: (1) Budget constraint: P t C t + Q t B t B t 1 + W t N t T t (2) (and a No-Ponzi Game constraint ruling out explosive debt) Household chooses optimal paths of C t, N t and B t (taking prices P t, W t and Q t as given) Let s solve its maximization problem using a Lagrangian method 3

5 @L = 0; U c;t t P t = 0; U c;t = t P t = 0; U n;t + t W t = 0; U n;t = t W t = 0; t Q t E t f t+1 g = 0; t Q t = E t f t+1 g t From (*) and (**): From (*) and (***) With U (C t ; N t ) C1 t 1 N ' t 1+', ; ' > 0, U n;t U c;t = W t P t (4) Q t = E t Uc;t+1 U c;t P t P t+1 (5) C N ' t Q t = W t (6) P t ( Ct+1 ) P t = E t (7) C t P t+1 4

6 Log-linear versions of households optimality conditions (lower-case letters are logs or log-deviations from steady state) Labor supply: c t + 'n t = w t p t (8) Bonds pay out 1 the period after the purchase at price Q t. Bond yield is: This is the nominal interest rate yield = 1 Q t Q t 1 + yield = Qt 1 log (1 + yield) = log Q t yield ' i t With this de nition, consumption-euler equation becomes: where t+1 p t+1 p t and log Money demand is postulated (for now): log Q t c t = E t fc t+1 g 1 (i t E t f t+1 g ) (9) m t p t = y t i t ; > 0 (10) 5

7 Firms behavior Production function for the representative rm Y t = A t Nt 1 ; 0 < < 1 (11) Pro ts in each period t: P t Y t W t N t (12) Firms choose N t to maximize P t A t N 1 t W t N t First-order condition (1 ) P t A t N t = W t ; (1 ) A t N t = W t P t (13) Labor demand in logs: log (1 ) + a t n t = w t p t (14) 6

8 Equilibrium outcomes Goods market clearing (or, resource constraint, or, national account): y t = c t (*) Labor market equilibrium (labor supply equals labor demand): c t + 'n t = log (1 ) + a t n t (**) Bond market clearing (household s intertemporal consumption decision with goods market clearing) y t = E t fy t+1 g 1 (i t E t f t+1 g ) (***) Production function in the aggregate: y t = a t + (1 ) n t (****) (*), (**) and (****) gives c t, n t and y t (***) gives subsequently the real interest rate r t i t E t f t+1 g 7

9 Typical classical features: Real variables like consumption, output and employment (and thus the real wage) are determined from the economy s supply side No role for monetary policy here We haven t used the money demand function at all The real interest rate is determined independently of monetary factors Any uctuation in main macro variables will result from uctuations in a t We need not even say anything about the nominal interest rate or the price level If we want to determine nominal variables, we must specify monetary policy 8

10 Monetary policy and prices The nominal interest rate as policy instrument Most central bank uses the nominal interest rate as policy instrument Model Problem: Specifying some exogenous path for the nominal interest rate in the model will not determine the price level. Note the Fisherian relationship : E t f t+1 g = i t r t where r t is independent of monetary policy Hence, (a rational expectations di erence equation) E t fp t+1 g = p t + i t r t (*) We can now introduce a shock which has nothing to do with the economy, t+1 : E t f t+1 g = 0. Then any price level satisfying p t+1 = p t + i t r t + t+1 is consistent with (*). The price level is indeterminate Likewise the money supply: m t = p t + y t Likewise the nominal wage i t 9

11 An exogenous path for the nominal interest rate does not help determine nominal variables in the classical model The problem is that the exogeneity leaves the economy without a nominal anchor A solution is to specify the nominal interest rate as a feedback rule where it is speci ed to be adjusted in response to some nominal variable One possibility (out of many!) i t = + t ; > 0 From the Fisher equation: t = E t f t+1 g + r t = E t f t+1 g + br t (22) If > 1, we have a unique stationary (i.e., non-explosive) solution t = 1X k=0 (1+k) E t fbr t+k g (23) If < 1, we have that any path of t satisfying (24) is an equilibrium (so, we have indeterminacy) t+1 = t br t+k + t+1 (24) 10

12 Taylor (1993): The Taylor Rule: i t = + 1:5 ( t ) + 0:5y t Note the > 1 feature: This is the Taylor principle 11

13 The nominal money supply as policy instrument In this case the economy has a nominal anchor : Consider an exogenous path for m t Combine the money demand equation and the Fisher equation m t p t = y t i t i t = E t fp t+1 p t g + r t Hence, p t = p t = y t + (E t fp t+1 p t g + r t ) + m t 1 + E tp t m t + u t u t (r t y t ) (independent of monetary policy) Since > 0 we have a unique stationary (i.e., non-explosive) solution p t = 1 1X k E t fm t+kg + u 0 t u 0 t k=0 1X k E t fu t+kg (independent of monetary policy) 1 + k=0 12

14 In terms of expected future money growth 1X p t = m t + (where is rst di erence operator) k=1 1 + k E t fm t+kg + u 0 t (25) For given expected money growth, prices move one for one with the money supply The nominal interest rate is found by inserting (25) into the money demand function i t = 1 (y t m t + p t ) X 1 k = 1 E t fm t+kg + u 00 t 1 + u 00 t k=1 1 (y t + u 0 t) (independent of monetary policy) The nominal interest rate increases with expected future money growth 13

15 An example of a process for nominal money growth Let m t = m m t 1 + " m t Assume no real shocks (hence y t and r t does not move; assume they are zero) Solution for prices and nominal interest rate p t = m t + m 1 + (1 m ) m t; i t = m 1 + (1 m ) m t If money growth is persistent there are large e ects on prices and positive e ect on interest rates No liquidity e ect; i.e., an expansionary money growth shock increases the nominal interest rate (and real money falls equivalently) 14

16 Do all classical models leave monetary policy useless? No While monetary neutrality prevails, many models with exible prices can have e ects of monetary policy through variations in in ation and the nominal interest rate There will not be monetary superneutrality Galí presents a model where money enters the utility function U = U (C t ; M t ; N t ) Short cut for money s liquidity services (e.g., saved leisure) The model features a micro-founded money demand relation just as postulated before Labor supply can now be a ected by changes in the nominal interest rate: U n;t U c;t = W t P t is a ected if marginal rate of substitution is a ected by money 15

17 Dynamic e ects of money shocks in a classical money-in-the utility function (from Walsh, 2010) To assess the quantitative e ects of money shocks, a money-in-the-utility function model is calibrated and simulated Calibration: Assign empirically plausible values to the parameters of the model Simulation: Perform a linearization of the model s dynamic equations (everything is expressed as percentage deviations from steady state); solve this system by numerical methods (various simulation programs are available on the internet); create arti cial time series data from the system From the arti cial data, one evaluates the properties of the model in terms of: Standard deviations of various relevant variables, and their s.d. relative to output Correlation coe cients of various variables with output Impulse response patterns of variables when shocks hit 16

18 Main results (when U cm > 0) Steady-state non-superneutrality is of the form of: Higher money growth =) higher in ation and nominal interest rate =) lower money demand =) lower marginal utility of consumption =) lower labor supply =) output If money growth shocks, " m t, shall pay a role, persistence in money growth is necessary ( m > 0 is needed). Then, the shock will a ects expected next-period in ation, and thus through the Fisher equation period t nominal interest rate. The e ects of money shocks on labor and output are stronger the more persistence in money growth, but the e ects are quantitatively very small (nothing compared to what we saw from VARs) Main e ects of money shocks are on in ation and nominal interest rates Positive money shocks lead to higher nominal interest rates. In contrast with liquidity e ect seen in VARs (and in contrast with usual IS/LM story where nominal rates fall to increase money demand). Reason is exible prices: Prices adjust instantaneously so as to reduce real money supply, matching the fall in demand resulting from higher nominal interest rates. 17

19 Source: Walsh, (2010): Monetary Theory and Policy (The MIT Press) ( u = m ) 18

20 Source: Walsh, (2010): Monetary Theory and Policy (The MIT Press) ( u = m ) 19

21 Summary The classical model has little role for money in determination of real variables It should be seen as a model of the long run, or, the average hypothetical evolution of the economy I.e., it is important as a benchmark for showing the e cient allocation in a micro-founded economy (i.e., an ideal, but unrealistic, world) Some, like the money-in-the-utility function model can be used to assess the optimal long-run in ation rate Optimum: U m = 0. This requires i = 0 and thus = The Friedman rule; private opportunity cost of money is equal to social costs (zero) r Model is not suitable for analyzing the short run implications of monetary shocks as the models, by nature, exhibits monetary neutrality (although not necessarily superneutrality). Impulse response patterns are unrealistic, even when there are e ects on real variables Short and long run are virtually indistinguishable To remedy the short-run failure of such models, one must introduce nominal rigidities; This is the aim of New Keynesian Theory 20

22 Next time(s) Monday, November 22: Exercises: Derive equation (9) in Galí, Chapter 2 Derive equations (17), (18), (19) and (20) and interpret economically how the variables are a ected by a t. Derive (23) and (24) and interpret the role of for uniqueness of equilibrium Derive the solutions for in ation, the nominal interest rate and output in the model with money-inthe utility function on page 31 in Galí, Chapter. Interpret economically the e ects of a t and t.(this requires deriving the model from the beginning!) Tuesday, November 23 Lecture: The Basic New Keynesian Model (Galí, Chapter 3) 21